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Chapter 5: Problem 15

Solve. $$\frac{d y}{d x}=4 x^{3} y$$

Short Answer

Expert verified
The solution is \(y = Ke^{x^4}\), where \(K\) is a constant.

Step by step solution

01

- Separate variables

Rewrite the given differential equation by separating the variables. We have \[\frac{dy}{dx} = 4x^3 y\]. To separate variables, divide both sides by y and multiply both sides by dx: \[\frac{1}{y} dy = 4x^3 dx\].
02

- Integrate both sides

Integrate both sides to solve for y. \[\int \frac{1}{y} \, dy = \int 4x^3 \, dx\].
03

- Perform the integration

Carry out the integrations on both sides. The left side integrates to \[\ln|y|\], and the right side integrates to \[x^4\]. So we get: \[\ln|y| = x^4 + C\], where C is the constant of integration.
04

- Solve for y

Exponentiate both sides to solve for y. \[e^{\ln|y|} = e^{x^4 + C}\]. This simplifies to \[|y| = e^{x^4} e^C\]. Let \[e^C = K\], where K is a new constant. So, \[|y| = K e^{x^4}\].
05

- Remove absolute value

Assume \(y\) can be positive or negative, the solution becomes \[y = Ke^{x^4}\], where \(K\) can be any constant.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

separation of variables
Separation of variables is a technique used to solve differential equations. The main idea is to rearrange the equation so that each variable and its derivative are on opposite sides. In the given problem, we started with \[\frac{d y}{d x} = 4 x^{3} y\]. By dividing both sides by \y\ and multiplying by \dx\, we obtained \[\frac{1}{y} dy = 4x^3 dx\]. This effectively separates the variables, with all \y\ components on the left and all \x\ components on the right. Separated variables make it easier to solve the equation through integration.
integration
Integration is a fundamental concept in calculus, where we find the antiderivative or the area under a curve. In our separated equation \[\frac{1}{y} dy = 4x^3 dx\], we need to integrate both sides to find \y\. The integral of \[\frac{1}{y} dy\] is \[\text{ln}|y|\], and the integral of \[4x^3 dx\] is \[x^4\]. Thus, we obtain \[\text{ln}|y| = x^4 + C\].
constant of integration
The constant of integration, represented by \C\, appears when we integrate an expression. It accounts for any constant that could have been differentiated to zero when finding the original function. In our solution, after integrating, we had \[\text{ln}|y| = x^4 + C\]. This \C\ is important because it influences the final value of the solution, maintaining the generality of the equation.
exponential function
The exponential function, denoted as \[e^{x}\], is a crucial part of solving our differential equation. To isolate \y\, we exponentiated both sides of \[ \text{ln} |y| = x^4 + C \]. This gave \[e^{\text{ln} |y|} = e^{x^4 + C} \]. From the properties of logarithms and exponentials, we simplified this to \[ |y| = e^{x^4 + C} \] or \[ |y| = e^C e^{x^4} \]. By defining \[e^C\] as a new constant \K\, we found the solution \y = \pm K e^{x^4}\.

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Most popular questions from this chapter

Negotiating a sports contract. Gusto Stick is an excellent professional baseball player who has just become a free agent. His attorney begins negotiations with an interested team by asking for a contract that provides Gusto with an income stream given by \(R_{1}(t)=800,000+340,000 t\) over 10 yr, where \(t\) is in years. (Round all answers to the nearest \(\$ 100 .)\) a) What is the accumulated future value of the offer, assuming an interest rate of \(8 \%,\) compounded continuously? b) What is the accumulated present value of the offer, assuming an interest rate of \(8 \%,\) compounded continuously? c) The team counters by offering an income stream given by \(R_{2}(t)=600,000+210,000 t .\) What is the accumulated present value of this counteroffer? d) Gusto comes back with a demand for an income stream given by \(R_{3}(t)=1,000,000+250,000 t\) What is the accumulated present value of this income stream? e) Gusto signs a contract for the income stream in part (d) but decides to live on \(\$ 500,000\) each year, investing the rest at \(8 \%,\) compounded continuously. What is the accumulated future value of the remaining income, assuming an interest rate of \(8 \%,\) compounded continuously?

Solve. $$\frac{d y}{d x}=5 x^{4} y$$

Find the volume generated by rotating about the \(x\) -axis the regions bounded by the graphs of each set of equations. $$y=\frac{1}{x}, x=1, x=4$$

Radioactive implant treatments. In the treatment of prostate cancer, radioactive implants are often used. The implants are left in the patient and never removed. The amount of energy that is transmitted to the body from the implant is measured in rem units and is given by $$ E=\int_{0}^{a} P_{0} e^{-k t} d t$$ where \(k\) is the decay constant for the radioactive material, a is the number of years since the implant, and \(P_{0}\) is the initial rate at which energy is transmitted. Use this information for Exercises 41 and 42 Suppose that the treatment uses palladium-103, which has a half-life of 16.99 days. a) Find the decay rate, \(k,\) of palladium- 103 b) How much energy (measured in rems) is transmitted in the first month if the initial rate of transmission is 10 rems per year? c) What is the total amount of energy that the implant will transmit to the body?

Find the particular solution determined by the given condition. Show that \(y=-2 e^{x}+x e^{x}\) is a solution of $$y^{\prime \prime}-2 y^{\prime}+y=0$$.
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